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On cubics and quartics through a canonical curve Christian Pauly October

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Niveau: Supérieur, Doctorat, Bac+8
On cubics and quartics through a canonical curve Christian Pauly October 21, 2003 Abstract We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve. 2000 Mathematics Subject Classification: 14H60, 14H42. 1 Introduction Let C be a smooth nonhyperelliptic curve of genus g ≥ 4 defined over the complex numbers, which we consider as an embedded curve ?? : C ?? Pg?1 by its canonical linear series |?|. Let I = ?n≥2 I(n) be the graded ideal of the canonical curve. It was classically known (Noether- Enriques-Petri theorem, see e.g. [ACGH] p. 124) that the ideal I is generated by its elements of degree 2, unless C is trigonal or a plane quintic. It was also classically known how to construct some distinguished quadrics in I(2). We consider a double point of the theta divisor ? ? Picg?1(C), which corresponds by Riemann's singularity theorem to a degree g ? 1 line bundle L satisfying dim |L| = dim |?L?1| = 1 and we observe that the morphism ?L? ??L?1 : C ?? C ? ? |L|?? |?L?1|? = P1?P1 (here

  • rational map

  • ?? e?w ??

  • tangent space

  • canonical curve

  • vector bundle

  • also apply

  • vector bundles over

  • m2 ??


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OncubicsandquarticsthroughacanonicalcurveChristianPaulyOctober21,2003AbstractWeconstructfamiliesofquarticandcubichypersurfacesthroughacanonicalcurve,whichareparametrizedbyanopensubsetinaGrassmannianandaFlagvarietyrespectively.UsingG.Kempf’scohomologicalobstructiontheory,weshowthatthesefamiliescutoutthecanonicalcurveandthatthequarticsarebirational(viaablowing-upofalinearsubspace)toquadricbundlesovertheprojectiveplane,whoseSteineriancurveequalsthecanonicalcurve.2000MathematicsSubjectClassification:14H60,14H42.1IntroductionLetCbeasmoothnonhyperellipticcurveofgenusg4definedoverthecomplexnumbers,whichLweconsiderasanembeddedcurveιω:C,Pg1byitscanonicallinearseries|ω|.LetI=n2I(n)bethegradedidealofthecanonicalcurve.Itwasclassicallyknown(Noether-Enriques-Petritheorem,seee.g.[ACGH]p.124)thattheidealIisgeneratedbyitselementsofdegree2,unlessCistrigonaloraplanequintic.ItwasalsoclassicallyknownhowtoconstructsomedistinguishedquadricsinI(2).WeconsideradoublepointofthethetadivisorΘPicg1(C),whichcorrespondsbyRiemann’ssingularitytheoremtoadegreeg1linebundleLsatisfyingdim|L|=dim|ωL1|=1andweobservethatthemorphismιL×ιωL1:C−→C0⊂|L|×|ωL1|=P1×P1(hereC0denotestheimagecurve)followedbytheSegreembeddingintoP3factorizesthroughthecanonicalspace|ω|,i.e.,C,→|ω|πP1×P1,P3,whereπisprojectionfroma(g5)-dimensionalvertexPVin|ω|.WethendefinethequadricQL:=π1(P1×P1),whichisarank4quadricinI(2)andcoincideswiththeprojectivizedtangentconeatthedoublepoint[L]ΘundertheidentificationofH0(C,ω)withthetangentspaceT[L]Picg1(C).Themainresult,duetoM.Green[Gr],assertsthatthesetofquadrics{QL},whenLvariesoverthedoublepointsofΘ,linearlyspansI(2).FromthisresultoneinfersaconstructiveTorellitheorembyintersectingallquadricsQL—atleastforCgeneralenough.ThegeometryofthethetadivisorΘatadoublepoint[L]canalsobeexploitedtoproducehigherdegreeelementsintheidealIasfollows:weexpandinasuitablesetofcoordinatesalocalequationθofΘnear[L]asθ=θ2+θ3+...,whereθiarehomogeneousformsofdegreei.HavingseenthatQL=Zeros(θ2),wedenotebySLthecubicZeros(θ3)⊂|ω|,theosculatingconeof1